3.39 \(\int x^4 \text{PolyLog}(3,a x^2) \, dx\)

Optimal. Leaf size=87 \[ -\frac{2}{25} x^5 \text{PolyLog}\left (2,a x^2\right )+\frac{1}{5} x^5 \text{PolyLog}\left (3,a x^2\right )+\frac{8 x}{125 a^2}-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{125 a^{5/2}}+\frac{8 x^3}{375 a}-\frac{4}{125} x^5 \log \left (1-a x^2\right )+\frac{8 x^5}{625} \]

[Out]

(8*x)/(125*a^2) + (8*x^3)/(375*a) + (8*x^5)/625 - (8*ArcTanh[Sqrt[a]*x])/(125*a^(5/2)) - (4*x^5*Log[1 - a*x^2]
)/125 - (2*x^5*PolyLog[2, a*x^2])/25 + (x^5*PolyLog[3, a*x^2])/5

________________________________________________________________________________________

Rubi [A]  time = 0.0530276, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2455, 302, 206} \[ -\frac{2}{25} x^5 \text{PolyLog}\left (2,a x^2\right )+\frac{1}{5} x^5 \text{PolyLog}\left (3,a x^2\right )+\frac{8 x}{125 a^2}-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{125 a^{5/2}}+\frac{8 x^3}{375 a}-\frac{4}{125} x^5 \log \left (1-a x^2\right )+\frac{8 x^5}{625} \]

Antiderivative was successfully verified.

[In]

Int[x^4*PolyLog[3, a*x^2],x]

[Out]

(8*x)/(125*a^2) + (8*x^3)/(375*a) + (8*x^5)/625 - (8*ArcTanh[Sqrt[a]*x])/(125*a^(5/2)) - (4*x^5*Log[1 - a*x^2]
)/125 - (2*x^5*PolyLog[2, a*x^2])/25 + (x^5*PolyLog[3, a*x^2])/5

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^4 \text{Li}_3\left (a x^2\right ) \, dx &=\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )-\frac{2}{5} \int x^4 \text{Li}_2\left (a x^2\right ) \, dx\\ &=-\frac{2}{25} x^5 \text{Li}_2\left (a x^2\right )+\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )-\frac{4}{25} \int x^4 \log \left (1-a x^2\right ) \, dx\\ &=-\frac{4}{125} x^5 \log \left (1-a x^2\right )-\frac{2}{25} x^5 \text{Li}_2\left (a x^2\right )+\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )-\frac{1}{125} (8 a) \int \frac{x^6}{1-a x^2} \, dx\\ &=-\frac{4}{125} x^5 \log \left (1-a x^2\right )-\frac{2}{25} x^5 \text{Li}_2\left (a x^2\right )+\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )-\frac{1}{125} (8 a) \int \left (-\frac{1}{a^3}-\frac{x^2}{a^2}-\frac{x^4}{a}+\frac{1}{a^3 \left (1-a x^2\right )}\right ) \, dx\\ &=\frac{8 x}{125 a^2}+\frac{8 x^3}{375 a}+\frac{8 x^5}{625}-\frac{4}{125} x^5 \log \left (1-a x^2\right )-\frac{2}{25} x^5 \text{Li}_2\left (a x^2\right )+\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )-\frac{8 \int \frac{1}{1-a x^2} \, dx}{125 a^2}\\ &=\frac{8 x}{125 a^2}+\frac{8 x^3}{375 a}+\frac{8 x^5}{625}-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{125 a^{5/2}}-\frac{4}{125} x^5 \log \left (1-a x^2\right )-\frac{2}{25} x^5 \text{Li}_2\left (a x^2\right )+\frac{1}{5} x^5 \text{Li}_3\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.161049, size = 77, normalized size = 0.89 \[ \frac{-150 x^5 \text{PolyLog}\left (2,a x^2\right )+375 x^5 \text{PolyLog}\left (3,a x^2\right )+\frac{120 x}{a^2}-\frac{120 \tanh ^{-1}\left (\sqrt{a} x\right )}{a^{5/2}}+\frac{40 x^3}{a}-60 x^5 \log \left (1-a x^2\right )+24 x^5}{1875} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*PolyLog[3, a*x^2],x]

[Out]

((120*x)/a^2 + (40*x^3)/a + 24*x^5 - (120*ArcTanh[Sqrt[a]*x])/a^(5/2) - 60*x^5*Log[1 - a*x^2] - 150*x^5*PolyLo
g[2, a*x^2] + 375*x^5*PolyLog[3, a*x^2])/1875

________________________________________________________________________________________

Maple [B]  time = 0.178, size = 144, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,{a}^{2}} \left ({\frac{2\,x \left ( 168\,{a}^{2}{x}^{4}+280\,a{x}^{2}+840 \right ) }{13125\,{a}^{3}} \left ( -a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,x}{125\,{a}^{3}} \left ( -a \right ) ^{{\frac{7}{2}}} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ){\frac{1}{\sqrt{a{x}^{2}}}}}-{\frac{8\,{x}^{5}\ln \left ( -a{x}^{2}+1 \right ) }{125\,a} \left ( -a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{x}^{5}{\it polylog} \left ( 2,a{x}^{2} \right ) }{25\,a} \left ( -a \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{x}^{5}{\it polylog} \left ( 3,a{x}^{2} \right ) }{5\,a} \left ( -a \right ) ^{{\frac{7}{2}}}} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*polylog(3,a*x^2),x)

[Out]

-1/2/a^2/(-a)^(1/2)*(2/13125*x*(-a)^(7/2)*(168*a^2*x^4+280*a*x^2+840)/a^3+8/125*x*(-a)^(7/2)/a^3/(a*x^2)^(1/2)
*(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))-8/125*x^5*(-a)^(7/2)/a*ln(-a*x^2+1)-4/25*x^5*(-a)^(7/2)*polylog(2,a
*x^2)/a+2/5*x^5*(-a)^(7/2)/a*polylog(3,a*x^2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 2.77746, size = 608, normalized size = 6.99 \begin{align*} \left [-\frac{150 \, a^{3} x^{5}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) + 60 \, a^{3} x^{5} \log \left (-a x^{2} + 1\right ) - 375 \, a^{3} x^{5}{\rm polylog}\left (3, a x^{2}\right ) - 24 \, a^{3} x^{5} - 40 \, a^{2} x^{3} - 120 \, a x - 60 \, \sqrt{a} \log \left (\frac{a x^{2} - 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right )}{1875 \, a^{3}}, -\frac{150 \, a^{3} x^{5}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) + 60 \, a^{3} x^{5} \log \left (-a x^{2} + 1\right ) - 375 \, a^{3} x^{5}{\rm polylog}\left (3, a x^{2}\right ) - 24 \, a^{3} x^{5} - 40 \, a^{2} x^{3} - 120 \, a x - 120 \, \sqrt{-a} \arctan \left (\sqrt{-a} x\right )}{1875 \, a^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*polylog(3,a*x^2),x, algorithm="fricas")

[Out]

[-1/1875*(150*a^3*x^5*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) + 60*a^3*x^5*log(-a*x^2 + 1) - 375
*a^3*x^5*polylog(3, a*x^2) - 24*a^3*x^5 - 40*a^2*x^3 - 120*a*x - 60*sqrt(a)*log((a*x^2 - 2*sqrt(a)*x + 1)/(a*x
^2 - 1)))/a^3, -1/1875*(150*a^3*x^5*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) + 60*a^3*x^5*log(-a*
x^2 + 1) - 375*a^3*x^5*polylog(3, a*x^2) - 24*a^3*x^5 - 40*a^2*x^3 - 120*a*x - 120*sqrt(-a)*arctan(sqrt(-a)*x)
)/a^3]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{Li}_{3}\left (a x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*polylog(3,a*x**2),x)

[Out]

Integral(x**4*polylog(3, a*x**2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\rm Li}_{3}(a x^{2})\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*polylog(3,a*x^2),x, algorithm="giac")

[Out]

integrate(x^4*polylog(3, a*x^2), x)